Design of Vibration Absorbers

Objective:

To reduce the vibration of a

primary device by adding an absorber to the system

Applications:

• Reciprocating machines

• Building excited by an earthquake

• Transmission lines or telephone lines excited by wind blowing

Applications (1)

Vibration absorber in the transmission lines

Tuned mass dampers beneath the bridge platform.

Applications (2)

Tuned Mass Damper

in the building

How vibration absorber works

Vibration absorber is applied to the machine whose operation frequency meets its resonance frequency.

m

ma

m 1dof

2dof

Frequency

Dis

p. (m

ain

mass)

Operation freq.

Vibration absorber is often used with

machines run at constant speed or

systems with const. excited freq., because

the combined system has narrow operating bandwidth.

Principle (1)

EOM

0

sin

0

0 0 tF

x

x

kk

kkk

x

x

m

m

aaa

aa

aa

Synchronous motion

tXtx sin)(

tXtx aa sin)( Sub. Into EOM

tF

tX

X

mkk

kmkk

aaaa

aa

sin

0sin

0

2

2

0

0

1

2

2 F

mkk

kmkk

X

X

aaa

aa

a

Principle (2)

0

0

1

2

2 F

mkk

kmkk

X

X

aaa

aa

a

0

0

20

2

2 )(1

0

1

Fk

FmkF

mkkk

kmk

X

X

a

aa

aa

aaa

a

222 ))(( aaaa kmkmkk where

0

2 )( FmkX aa

0Fk

X aa

Principle (3)

0

2 )( FmkX aa

0Fk

X aa

ma and ka can be chosen such that X = 0

a

a

m

k2

tk

Ftx

a

a sin)( 0Motion of absorber mass:a

ak

FX 0;

Force acting on the absorber mass: 00 )/( FkFkxk aaaa

Force provided by ma = Disturbance force

Zero net force acting on the primary mass

Principle (4)

222

0

2

))((

)(

aaaa

aa

kmkmkk

FmkX

From

Define

a

aa

m

k

m

kp Original natural freq. of the primary

system without the absorber

Original natural freq. of the absorber before it is attached to the primary system

Normalize parametersm

map

a

a

r

Normalize disp. of the primary mass

2222

2

0 )1)(1(

1

rr

r

F

Xk

Principle (5)

2222

2

0 )1)(1(

1

rr

r

F

Xk

• Shade are is the useful operating bandwidth

(0.908a < < 1.118a)

• ma and ka are chosen such that r is within the bandwidth

• When r = 0.781 or 1.28, the combined system will experience the resonance and fail

Bandwidth of operating frequency

m

map

a

• As is increased, n split farther apart, and farther from the

operating point a

• 0.05 < < 0.25 (recommend)

• Very large large ma stress and fatigue problems

01)]1(1[2

22

2

2

22

a

n

a

n

Damping in vibration absorption

• Damping can reduce the resonance amplitude of the system

• Amplitude at operating point increase with increasing damping

Design procedure

• Select which will be tuned to zero amplitude

• Relation between ka and ma is obtained from 2 = ka/ma

• Select ma and ka (consider restrictions: force, motion of

absorber mass)

• Check the ratio = ma/m (recommended value: 0.05<<0.25)

Example 5.3.1

A radial saw base has a mass of 73.16 kg and is driven by a

motor. The motor runs at constant speed and produces a

13-N force at 180 cpm. The manufacturer wants a vibration

absorber designed to drive the table oscillation to zero.

Design the absorber assuming that the stiffness provided by

the table legs is 2600 N/m. Absorber has a maximum deflection of 0.2 cm.

Example

A diesel engine, weighting 3000 N, is supported on a

pedestal mount. The engine induces vibration through its

pedestal mount at an operating speed of 6000 rpm.

Determine the parameters of the vibration absorber what will

reduce the vibration when mounted on the pedestal. The

magnitude of the exciting force is 250 N, and the amplitude of motion of the absorber mass is to limited to 2 mm.

Example

A pipe carrying steam through a section of a factory vibrates

violently when the driving pump hits a speed of 232 rpm. In

an attempt to design an absorber, a trial 1 kg absorber tuned

to 232 rpm was attached. By changing the pump speed, it

was found that the pipe-absorber system has a resonance at

198 rpm. Redesign the absorber so that the natural frequencies are less than 160 rpm and more than 320 rpm.